The wave equation has been the object of intense study in the past decade. For a summary of the subject we refer the reader to [1]}, [2]} and for sharp decay results we refer to [3]}, [4]}, [5]}, [6]}). The first attempt to resolve Price's conjecture is due to Donninger, Schlag and Soffer [7]} where the \(2\ell +3\)  decay rate is proved for static initial data supported on the \(\ell \)  frequency (it is worth pointing out that for such data \(2\ell + 4\)  is the expected optimal decay rate). Hintz [8]} recently derived the \(2\ell +3\)  upper bound for general initial data supported on the \(\ell \)  frequency on Schwarzschild backgrounds. Ma [9]} also derived almost sharp decay rates for \(\ell =1\)  angular modes of Maxwell fields.
